Abstract
In this paper, we consider some paraconsistent calculi in a Hilbert-style formulation with the rule of detachment as the sole rule of interference. Each calculus will be expected to contain all axiom schemas of the positive fragment of classical propositional calculus and respect the principle of gentle explosion.
Highlights
The principle of explosion states that from any set {α, ¬α} of contradictory formulas any other formula β follows
Paraconsistent logic can be described as a logic in which the principle does not hold
We examine several paraconsistent calculi that respect the so-called principle of gentle explosion, according to which from any set {α, ¬α, ¬¬α} of formulas any other formula β follows
Summary
The principle of explosion states that from any set {α, ¬α} of contradictory formulas any other formula β follows. We examine several paraconsistent calculi that respect the so-called principle of gentle explosion, according to which from any set {α, ¬α, ¬¬α} of formulas any other formula β follows. The set F of formulas is defined in the standard way using propositional variables from var and the symbols ¬,. Each calculus C discussed in this paper is expected to have all axiom schemas of the positive fragment of classical propositional calculus (CPC + , for short), that is, all instances of the following schemas:. For any Γ ⊆ F and α, β ∈ F : Γ ∪ {α} `C β iff ΓC α → β It follows from (A9), the deduction theorem and (MP) that the following lemma holds as well: Lemma 2. The formulas will be useful for proving the results presented below
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